AAWSAP DIRD, An Introduction to the Statistical Drake Equation, March 2010

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UNCLASSIFIED/ /fOR OFFI&IAk Wlii QPU,¥
ourselves to the simpler uniform distribution instead,
as shown in the nest section.
4. STEP 4: ASSUMING THE EASIEST
PUT DISTRIBUTION FOR EACH D; :
THE U IFORM DISTRIBUTION
Let us now suppose that each of the seven D; is
distributed UNIFORMLY in the interval ranging
from the lower limit a; ~ 0 to the upper limit
b; ~a;·
This is the same as saying that the probability
density function of each of the seven Drake random
variables D; has the equation
1
funi rom, o. (x) = -
-
with O::; a; s; x s; b;
(10)
- '
h;-a;
as it follows at once from the normalization condition
bf , funi rol'TlLD-(x )dx = 1.
(11)
ti;
I
Let us now consider the mean value of such
uniform D; defined by
b;
1
f b'
(unifom'LD;)= f xfunibniLo, (x)dx=---
xdx
ai
b; -a; a;
2 ]b;
2
2
= --1-
.::._
= b; - a;
= a; +b;
[
b; - a;
2
a;
2 (b; - a;)
2
•
By words (as it is intuitively obvious): the mean
value of the uniform distribution simply is the mean
of the lower plus upper limit of the variable range
a -+b.
(unifonn_D; ) = -'--' .
(12)
2
In order to find the variance of the uniform
distribution, we first need finding the second moment
•
2)
fb; 2
( )
(uniform_Di
=
X funibrnLD, X dx
0 1
I
", ,
l
x 3 ]b;
h-3 - a. 3
[ 3
= b-a . Lx-dx = b. -a .
= 3(b. -;, .)
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<l;
I
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_ (b; -a;)(a; +a;b; +b; ) _ a; +a;b; +b;
-
3 (b; - a;)
-
3
The second moment of the uniform distribution is
thus
D 2)
a; +a;b; +b;
( unitiorm
.
= ---'---'-'-----'-
(13)
-
I
3
From (12 and (13) we may now derive the variance
of the uniform distribution
c:r~nifi>rnLD; = ( uniform_D ; 2)-(uniform_D;) 2
a;
? + a;b; +b;
2
=
( 14)
3
4
12
Upon taking the square root of both sides of ( 14 ), we
finally obtain the sta11dard deviation of the u11iform
distribution :
We now wish to perform a calculation that is
mathematically trivial, but rather unexpected from
the intuitive point of view, and very important for our
applications to the statistical Drake equation. Just
consider the two simultaneous equations (12) and
(15)
a. +b­
)
uniform_Di =~
l(
(16)
b-- a
(T
__
, __,
uniIDTITLD; -
2✓
3 •
Upon inverting this trivial linear system, one finds
fa; = (uniform_D; )- ✓3 (Tuni DnTLD;
(17)
lb; = ( uniform_D;) + ✓3 c:runifom1..D, •
This is of paramount importance for our application
the Statistical Drake equation inasmuch as it shows
that:
if one (scientifically) assigns the mean value and
standard deviation of a certai11 Drake random
variable D;, then. the lower a11d upper limits of the
relevant uniform distribution are given by the two
equations (17), respectively.
UNCLASSIFIED/ /POlt 9Pfl@IAL H§E 8Nk¥
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